Sound propagation in different media is central to architectural acoustics because the materials and environments surrounding a source determine how sound reaches listeners. Whether sound travels through air, solid walls, water, or porous absorbers, each medium shapes the wave's speed, direction, and energy loss in distinct ways.

This topic covers wave behavior in air, solids, liquids, porous materials, and multilayered structures. It also addresses diffraction, scattering, refraction, and diffusion, all of which influence how sound fields develop in real architectural spaces.

Sound propagation in air

Sound waves in air are the starting point for nearly every architectural acoustics problem. Room acoustics, noise transmission between spaces, and outdoor sound propagation all depend on how sound behaves as it moves through air.

Three things matter most: the speed of sound (which controls timing and wavelength), geometric spreading (which controls how quickly levels drop with distance), and air absorption (which selectively removes high-frequency energy over long paths).

Speed of sound in air

The speed of sound in air is approximately 343 m/s at 20°C and sea level. It increases by about 0.6 m/s for every 1°C rise in temperature.

Under normal conditions, air pressure and humidity have only minor effects on sound speed. The governing formula is:

$c = \sqrt{\frac{\gamma R T}{M}}$

where $\gamma$ is the adiabatic index (about 1.4 for air), $R$ is the universal gas constant, $T$ is the absolute temperature in Kelvin, and $M$ is the molar mass of air.

Factors affecting sound speed

Temperature is the dominant factor. Higher temperatures mean faster molecular motion and faster sound propagation. A temperature gradient across a space can bend (refract) sound waves.
Humidity has a small effect. More humid air is slightly less dense (water vapor is lighter than nitrogen and oxygen), so sound speed increases marginally.
Gas composition matters if the air mixture changes significantly, for example in spaces with elevated $CO_2$ concentrations.

Attenuation of sound in air

Sound intensity drops with distance for two reasons: geometric spreading and air absorption.

Geometric spreading is the primary cause of attenuation in open (free-field) conditions:

A point source loses 6 dB per doubling of distance (inverse square law)
A line source (like a busy highway) loses 3 dB per doubling of distance

Air absorption adds frequency-dependent losses on top of geometric spreading. It results from viscous friction, thermal conduction, and molecular relaxation of oxygen and nitrogen. Air absorption is most significant at high frequencies and over long distances, which is why distant thunder sounds like a low rumble with the high-frequency crack stripped away.

Air absorption coefficients

Air absorption coefficients quantify the sound energy lost per unit distance, expressed in dB/m or dB/km. They vary with frequency, temperature, humidity, and pressure, and can be calculated using the standardized procedure in ISO 9613-1.

Higher frequencies experience much greater absorption. At 20°C and 50% relative humidity, absorption at 8 kHz can be roughly 50 times greater than at 500 Hz. This is an important consideration for large rooms (concert halls, arenas) and any long-distance outdoor propagation problem.

Sound propagation in solids

Solid materials form the walls, floors, and structural elements of buildings. Understanding how sound travels through them is essential for designing sound insulation and controlling vibration paths.

Unlike air, solids can carry more than one type of wave, and sound generally travels much faster through them.

Longitudinal vs. transverse waves

Longitudinal waves (compression waves, P-waves): particles oscillate parallel to the direction the wave travels, creating alternating compression and rarefaction. These are the same type of wave that propagates in air.
Transverse waves (shear waves, S-waves): particles oscillate perpendicular to the wave direction, causing shearing deformation.

Solids support both wave types. Fluids (air, water) support only longitudinal waves because they cannot sustain shear stress.

Speed of sound in solids

Sound speed in solids depends on stiffness and density:

Longitudinal wave speed: $c_L = \sqrt{\frac{E}{\rho}}$ , where $E$ is Young's modulus and $\rho$ is density
Transverse wave speed: $c_T = \sqrt{\frac{G}{\rho}}$ , where $G$ is the shear modulus

Typical values are much higher than in air: roughly 5,000 m/s in steel and ~3,000–4,000 m/s in concrete for longitudinal waves.

Note on the longitudinal speed formula: The expression $c_L = \sqrt{E/\rho}$ applies to thin bars (one-dimensional propagation). In bulk solids, the longitudinal wave speed also involves Poisson's ratio and is higher than the bar speed. For this course, the thin-bar formula is the standard reference.

Factors affecting sound speed

Stiffness vs. density: Stiffer materials (higher moduli) produce higher sound speeds; denser materials produce lower sound speeds. Steel is both stiff and dense, but its very high modulus wins out, giving it a fast sound speed.
Temperature: Sound speed in solids generally increases slightly with temperature.
Anisotropy: In materials like wood, sound speed varies with direction because the grain structure makes the material stiffer along certain axes.

Attenuation of sound in solids

Attenuation in solids comes from viscous losses, thermal conduction, and scattering at grain boundaries or internal inhomogeneities. It generally increases with frequency and temperature and is expressed as an attenuation coefficient (dB/m) or a dimensionless loss factor.

Metals like steel have low loss factors (sound travels far with little damping), while materials like rubber and viscoelastic polymers have high loss factors (useful for vibration damping).

Transmission loss through solids

Transmission loss (TL) measures how much sound insulation a solid element provides, expressed in dB as a function of frequency.

TL depends on:

Material density and stiffness
Thickness of the element
Angle of incidence of the sound wave
Frequency (through phenomena like the mass law and the coincidence effect)

The mass law predicts that TL increases by about 6 dB for each doubling of surface mass or frequency. The coincidence effect causes a dip in TL at the frequency where the bending wavelength in the panel matches the projected wavelength of the incident sound. Standardized measurement methods (e.g., ISO 10140) are used to characterize TL in practice.

Speed of sound in air, Speed of sound - Wikipedia

Sound propagation in liquids

Liquids are relevant to architectural acoustics in specialized contexts (swimming pool halls, water features, hydraulic systems) and more broadly in underwater acoustics.

Speed of sound in liquids

Sound speed in a liquid depends on its bulk modulus and density:

$c = \sqrt{\frac{K}{\rho}}$

where $K$ is the bulk modulus and $\rho$ is the density.

Sound travels faster in liquids than in air but generally slower than in solids. In fresh water at 20°C, the speed is approximately 1,480 m/s. In oil, it's roughly 1,400 m/s.

Factors affecting sound speed

Temperature: In water, sound speed increases by about 3 m/s per °C.
Pressure: Sound speed increases by roughly 1.6 m/s for every 100 m increase in water depth.
Salinity: Each 1 PSU (Practical Salinity Unit) increase raises sound speed by about 1.3 m/s.

These three factors together create the characteristic sound speed profile in the ocean, which bends sound paths and creates phenomena like the SOFAR channel.

Attenuation of sound in liquids

Attenuation in liquids results from viscous losses, thermal conduction, and molecular relaxation. It increases with frequency but is much lower than attenuation in air at the same frequency, which is why sound can propagate over very long distances underwater.

Underwater acoustics applications

While not the core of architectural acoustics, underwater applications illustrate key propagation principles:

Sonar (Sound Navigation and Ranging): Active sonar emits pulses and listens for echoes; passive sonar listens for sounds emitted by targets.
Underwater communication: Acoustic modems transmit data between submerged devices.
Seismic exploration: Sound pulses map subsurface geological structures.
Marine mammal monitoring: Passive acoustic arrays track whale and dolphin populations.

Sound propagation in porous materials

Porous materials like foams, mineral wool, and fiberglass are the workhorses of sound absorption in buildings. Sound enters the pore network, and friction and heat exchange convert acoustic energy into heat.

Porous material properties

Four properties govern how a porous material interacts with sound:

Porosity: The fraction of the material's volume that is open space (voids). Higher porosity generally means better absorption because more air can enter the material.
Flow resistivity: The resistance the material offers to airflow per unit thickness (units: Pa·s/m²). Too low and sound passes straight through; too high and sound reflects off the surface. There's an optimal range for effective absorption.
Tortuosity: A measure of how winding the internal pore paths are. Higher tortuosity forces sound to travel longer, more complex paths, increasing energy loss.
Pore size distribution: The range of pore diameters affects which frequencies are absorbed most effectively.

Sound absorption mechanisms

Viscous losses: Friction between oscillating air particles and the pore walls. This is the dominant mechanism at low frequencies and in materials with small pores.
Thermal losses: Heat exchange between compressed/rarefied air and the pore walls. More significant at high frequencies and in materials with larger pores.
Structural losses: Vibration of the material's solid skeleton. This is significant in elastic materials like flexible foams and fibrous batts.

Acoustic impedance of porous materials

The acoustic impedance of a porous material is the complex ratio of sound pressure to particle velocity at its surface. It determines how much sound is reflected versus absorbed.

Impedance depends on material properties, frequency, and angle of incidence. It can be measured directly using an impedance tube or predicted with models such as Delany-Bazley or Johnson-Champoux-Allard (JCA).

Modeling sound propagation in porous media

Three categories of models exist, each trading simplicity for accuracy:

Empirical models (e.g., Delany-Bazley, Miki): Based on curve-fitting to measured data. They require only flow resistivity as input, making them simple to use, but they're limited in accuracy outside the conditions they were fitted to.
Phenomenological models (e.g., Johnson-Champoux-Allard, Biot): Based on physical principles with measurable input parameters (porosity, tortuosity, flow resistivity, etc.). More accurate and versatile, but they require more detailed material characterization.
Numerical models (e.g., finite element method, boundary element method): Solve the governing equations directly on a computational mesh. Most accurate and flexible, but computationally expensive and require full material property data.

Sound propagation in multilayered media

Real building elements are rarely single materials. Walls, floors, and acoustic panels are typically multilayered assemblies (e.g., drywall + air gap + insulation + drywall). Predicting their acoustic performance requires understanding how sound reflects and transmits at each interface and how the layers interact.

Reflection and transmission at interfaces

When a sound wave hits the boundary between two media, some energy reflects and some transmits. The split depends on the acoustic impedance mismatch between the two materials.

For normal incidence (sound hitting the surface head-on):

Reflection coefficient: $R = \frac{Z_2 - Z_1}{Z_2 + Z_1}$
Transmission coefficient: $T = \frac{2Z_2}{Z_2 + Z_1}$

where $Z_1$ and $Z_2$ are the acoustic impedances of the two media.

The larger the impedance mismatch, the more energy is reflected. This is why an air-concrete interface reflects most sound energy: concrete's impedance is thousands of times greater than air's.

For oblique incidence, the coefficients also depend on the angle of incidence. Snell's law governs the relationship between incident and transmitted angles, and the wave may convert between longitudinal and transverse types at the interface.

Transfer matrix method

The transfer matrix method (TMM) is the standard analytical tool for multilayered systems. Here's how it works:

Represent each layer as a 2×2 transfer matrix that relates sound pressure and particle velocity at one face of the layer to the other face.
Multiply the matrices for all layers in sequence to get the overall system matrix.
Extract the reflection coefficient, transmission coefficient, or absorption coefficient from the overall matrix.

TMM is fast, accurate for laterally infinite flat layers, and widely used in commercial acoustic modeling software.

Modeling sound propagation in multilayered structures

Analytical models (transfer matrix method): Best for flat, homogeneous layers with simple geometries. Fast and accurate within their assumptions.
Numerical models (FEM, BEM): Handle complex geometries, inhomogeneous layers, and finite panel sizes. Computationally demanding.
Statistical energy analysis (SEA): A high-frequency approach that tracks average energy flow between coupled subsystems. Suited to complex structures with high modal densities, such as entire wall or floor assemblies at mid-to-high frequencies.

Applications in architectural acoustics

Sound insulation of walls and floors: Optimizing layer materials, thicknesses, and sequencing to maximize transmission loss. Adding an air gap or resilient channel between layers can dramatically improve performance by decoupling the two sides.
Acoustic panels and absorbers: Layering porous absorbers with air cavities and facing materials to achieve broadband absorption or target specific frequency ranges.
Noise barriers: Designing multilayered outdoor barriers to maximize insertion loss through a combination of mass, absorption, and diffraction control.

Diffraction and scattering of sound

Diffraction and scattering redistribute sound energy in ways that simple ray-based models don't predict. Both are critical for understanding how sound reaches shadowed areas and how surfaces shape the sound field.

Diffraction around obstacles

Diffraction occurs when a sound wave encounters an obstacle or opening with dimensions comparable to or smaller than its wavelength. The wave bends around the obstacle or spreads out after passing through the opening.

The Fresnel number quantifies the extent of diffraction: it's based on the ratio of the path length difference (over the barrier vs. direct) to the wavelength. A higher Fresnel number means more effective shielding.

Low frequencies diffract more readily than high frequencies (their wavelengths are longer relative to typical obstacles). This is why you can hear bass from behind a wall but the high frequencies are blocked.

Scattering from rough surfaces

Scattering occurs when a sound wave hits a surface with irregularities comparable to or larger than its wavelength. Instead of reflecting in a single direction (specular reflection), the energy spreads out in many directions.

The degree of scattering depends on the roughness parameter: the ratio of the surface irregularity size to the wavelength. More scattering produces a more diffuse sound field with fewer strong discrete reflections.

Scattering is deliberately introduced in concert halls and studios using shaped surfaces and diffusers to create a sense of envelopment and spatial uniformity.

Modeling diffraction and scattering effects

Analytical models: Geometrical Theory of Diffraction (GTD) and Uniform Theory of Diffraction (UTD) work well for simple geometries at high frequencies. The Kirchhoff approximation assumes obstacles are large and smooth relative to the wavelength.
Numerical models: Boundary Element Method (BEM) and Finite Difference Time Domain (FDTD) solve the wave equation directly and handle complex geometries at low to medium frequencies.
Hybrid models: Combine analytical and numerical approaches. For example, using GTD for the overall barrier geometry and BEM for a complex edge profile. This balances accuracy with computational cost.

Refraction and diffusion of sound

Refraction bends sound paths through non-uniform media, while diffusion spreads sound energy evenly through a space. Both shape the listening experience in architectural environments.

Refraction due to temperature gradients

When sound propagates through air with a non-uniform temperature distribution, its speed varies from point to point. Sound waves bend toward regions of lower temperature (where the speed of sound is slower).

On a warm day, the air near the ground is hotter than the air above, so sound bends upward and away from listeners, creating a "shadow zone." At night, the ground cools and the gradient reverses, bending sound downward and allowing it to carry farther. In large indoor spaces with temperature stratification (e.g., atria, industrial halls), refraction can redirect sound in unexpected ways.

Refraction due to wind gradients

Wind adds a velocity component to the effective speed of sound. Downwind, the effective speed is higher near the top of the gradient (where wind is faster), bending sound downward toward the ground. Upwind, the opposite occurs: sound bends upward, creating a shadow zone.

The extent of refraction depends on the magnitude and direction of the wind gradient. This is important for outdoor noise predictions (highways, airports, industrial sites) and for large ventilated indoor spaces with significant airflow.

Sound diffusion in rooms

Diffusion is the even distribution of sound energy throughout a space, both spatially and over time. A perfectly diffuse sound field has uniform sound pressure everywhere and equal energy traveling in all directions.

Diffusion is achieved through:

Irregular room shapes
Non-parallel surfaces (splayed walls, angled ceilings)
Diffusers: purpose-designed surface elements (e.g., Schroeder diffusers, QRD panels) that scatter reflections across a wide range of angles

High diffusion is desirable in concert halls, recording studios, and listening rooms because it creates a uniform listening experience, reduces the audibility of discrete echoes, and supports a smooth, enveloping reverberant field.